Homework Assignment 7:
(1) Let f(x) be the probability density function of the Pareto random variable; that is,
f(x)
=
αθ
α
/ (x+
θ
)
α
+ 1
,
α
> 0,
θ
> 0,x > 0.
Show that F(x) = 1 – [
θ
/ (x +
θ
)]
α
and
λ
(x) =
α
/ (x +
θ
).
(2) Use the hazard rate function to show that the Weibull distribution is light tailed when
α
> 1.
(Ans
λ
(x) =
αλ
x
α
1
.
Since the hazard rate function is increasing when
α
> 1, the distribution will have
a light tail.)
(3) An automobile collision insurance policy reimburses repair costs subject to a deductible of 250 per
accident.
In the event of an accident, repair costs can be modeled by a uniform random variable on the
interval (0,1500).
(a) Determine the expected value of the
claim payment per loss
random variable
.
(Ans 520.83)
(b) Determine the expected value of the
claim payment per payment
random variable. (Ans 625)
(c) Determine the expected value of the
claim payment per payment
random variable if the deductible
is eliminated. (Ans 750)
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 Spring '11
 CharlesDann
 Probability theory, insurance company, Weibull, 13.85%, 13.85 %

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